Fractional differentiation and Lipschitz spaces on local fields

Abstract:

In this paper we continue our study of differentiation on a local field K. We define strong derivatives of fractional order $\alpha > 0$ for functions in ${L_r}(\textbf {K})$, $1 \leqslant r < \infty$. After establishing a number of basic properties for such derivatives we prove that the spaces of Bessel potentials on K are equal to the spaces of strongly ${L_r}(\textbf {K})$-differentiable functions of order $\alpha > 0$ when $1 \leqslant r \leqslant 2$. We then focus our attention on the relationship between these spaces and the generalized Lipschitz spaces over K. Among others, we prove an inclusion theorem similar to a wellknown result of Taibleson for such spaces over ${\textbf {R}^n}$.